Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C4 |
| direct product | C5×#2245 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C5 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ C4×C5×C8 ◁ G | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,21,27)(16,23,31,42)(17,22,33,40)(18,25,35,37)(19,24,28,43)(20,36,34,26)(29,41,44,39)(30,45,32,38);(7,10)(11,13)(14,16,28,44,20,30,18,17,21,31,19,29,34,32,35,33)(15,22,37,45,26,39,24,23,27,40,25,38,36,41,43,42);(6,8,9,12)(7,11,10,13)(14,17,18,32,34,44,28,31,21,33,35,30,20,29,19,16)(15,23,24,41,36,45,37,40,27,42,43,39,26,38,25,22)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5835);